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  • English
Course Code: 
MATH 439
Semester: 
Fall
Course Type: 
Core
P: 
3
Lab: 
0
Laboratuvar Saati: 
0
Credits: 
3
ECTS: 
6
Course Language: 
English
Course Objectives: 
To develop the necessary background for modern analysis courses to follow
Course Content: 

Basic concepts about topological spaces and metric spaces. Complete metric spaces, Baire’s theorem, Contracting mapping theorem and its applications. Compact spaces,  Arzela-Ascoli  Theorem Seperability, second countability, Urysohn's lemma and the Tietze extension theorem, Connected spaces, Weierstrass approximation theorem

Course Methodology: 
1: Lecture, 2: Problem Solving
Course Evaluation Methods: 
A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes

Program Learning Outcomes

Teaching Methods

Assessment Methods

1) Learns basic concepts of topological spaces with emphasis on metric spaces

 

1,2

A

2) Learns Cauchy sequences and completeness

 

2,2

A

3) Learns the concept of compact space

 

1,2

A

4) Learns Baier’s category

 

1,2

A

5) Learns Ascoli-Arzela theorem,  Weierstrass approximation

 

1,2

A

6) Acquires the skill of applying these concepts

 

1,2

A

Course Flow

Week

Topics

Study Materials

1

Basic concepts about metric spaces and examples

 

2

Open, closed sets, topology and convergence

 

3

Cauchy sequences and complete metric spaces,  Baire's theorem

 

4

Continuity and uniformly continuity,  spaces of continuous functions, Euclidean space

 

5

Contracting mapping theorem and its applications

 

6

The definition of topological spaces and some examples , elementary concepts, Open bases and open subbases

 

7

Compact spaces, Products of spaces,Tychonoff's theorem and locally compact spaces

 

8

Compactness for metric spaces

 

9

Arzela-Ascoli  Theorem

 

10

Seperability, second countability

 

11

Hausdorff spaces, Completely regular spaces and normal spaces

 

12

Urysohn's lemma and the Tietze extension theorem

 

13

Connected spaces, The components of a space, Totally disconnected spaces, Locally connected spaces

 

14

The Weierstrass approximation theorem ,The Stone-Weierstrass theorems

 

Recommended Sources

Textbook

1. S. Kumaresan, Topology of Metric Spaces

2. George F. Simmons, Topology and Modern Analysis

3. W A Sutherland, Introduction to Metric and Topological Spaces

4. E T Copson, Metric Spaces

Additional Resources

 

Material Sharing

Documents

 

Assignments

 

Exams

 

Assessment

IN-TERM STUDIES

NUMBER

PERCENTAGE

Mid-terms

2

100

Quizzes

-

0

Assignments

-

0

Total

 

100

CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE

 

60

CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE

 

40

Total

 

100

 

COURSE CATEGORY

Expertise/Field Courses

Course’s Contribution to Program

No

Program Learning Outcomes

Contribution

1

2

3

4

5

1

The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)

       

X

2

Acquiring fundamental knowledge on fundamental research fields in mathematics

       

X

3

Ability form and interpret the relations between research topics in mathematics

       

X

4

Ability to define, formulate and solve mathematical problems

       

X

5

Consciousness of professional ethics and responsibilty

   

X

   

6

Ability to communicate actively

   

X

   

7

Ability of self-development in fields of interest

       

X

8

Ability to learn, choose and use necessary information technologies

X

       

9

Lifelong education

   

X

   

ECTS

Activities

Quantity

Duration
(Hour)

Total
Workload
(Hour)

Course Duration (14x Total course hours)

14

3

42

Hours for off-the-classroom study (Pre-study, practice)

14

4

56

Mid-terms (Including self study)

2

15

30

Quizzes

-

   

Assignments

-

   

Final examination (Including self study)

1

20

20

Total Work Load

 

 

148

Total Work Load / 25 (h)

 

 

5.92

ECTS Credit of the Course

 

 

6